Abstract

The purpose of this article is to introduce the concept of total quasi- ϕ-asymptotically nonexpansive mapping which contains many kinds of mappings as its special cases and we prove a strong convergence theorem by using a hybrid method for finding a common element of the set of solutions for a generalized mixed equilibrium problems, the set of fixed points of a family of total quasi- ϕ-asymptotically nonexpansive mappings in uniformly smooth and strictly convex Banach space with the Kadec-Klee property. The results presented in the article improve and extend some recent results.

MR (2000) Subject Classification: 47H09; 47H05.

Keywords

1 Introduction

Let E be a Banach space and C be a closed convex subsets of E. Let F be an equilibrium bifunction from C × C into R, ψ: C → R be a real-valued function and A: C → E* be a nonlinear mapping. The "so-called" generalized mixed equilibrium problem is to find z∈C such that

F(z,y)+〈Az,y-z〉+ψ(y)-ψ(z)≥0,∀y∈C.

(1.1)

The set of solutions of (1.1) is denoted by GMEP, i.e.,

GMEP={z∈C:F(z,y)+〈Az,y-z〉+ψ(y)-ψ(z)≥0,∀y∈C}.

Special examples:

(I)

If A = 0, then the problem (1.1) is equivalent to find z∈C such that

F(z,y)+ψ(y)-ψ(z)≥0,∀y∈C.

(1.2)

which is called the mixed equilibrium problem, see [1]. The set of solutions of (1.2) is denoted by MEP.

(II)

If F = 0, then the problem (1.1) is equivalent to find z∈C such that

〈Az,y-z〉+ψ(y)-ψ(z)≥0,∀y∈C.

(1.3)

which is called the mixed variational inequality of Browder type. The set of solutions of (1.3) is denoted by V I(C, A, ψ).

(III)

If ψ = 0, then the problem (1.1) is equivalent to find z∈C such that

F(z,y)+〈Az,y-z〉≥0,∀y∈C.

(1.4)

which is called the generalized equilibrium problem, see [2]. The set of solutions of (1.4) is denoted by EP.

(IV)

If A = 0, ψ = 0, then the problem (1.1) is equivalent to find z∈C such that

F(z,y)≥0,∀y∈C.

(1.5)

which is called the equilibrium problem. The set of solutions of (1.5) is denoted by EP(F).

These show that the problem (1.1) is very general in the sense that numerous problems in physics, optimization, and economics reduce to finding a solution of (1.1). Recently, some methods have been proposed for the generalized mixed equilibrium problem in Banach spaces (see, for example [1–7]).

Let E be a smooth, strictly convex, and reflexive Banach spaces and C be a nonempty closed convex subsets of E. Throughout this article, we denote by ϕ the function defined by

ϕ(y,x)=||y||2-2〈y,Jx〉+||x||2,∀x,y∈E,

(1.6)

where J: E → 2 E* is the normalized duality mapping.

Let T: C → C be a mapping and F (T) be the set of fixed points of T .

Recall that a point p∈C is said to be an asymptotic fixed point of T if there exists {xn } in C which converges weakly to p and limn→∞||xn - Txn || = 0. We denote the set of all asymptotic fixed point of T by F̃(T). A point p∈C is said to be a strong asymptotic fixed point of T if there exists {xn } in C such that xn → p and limn→∞||xn - Txn || = 0 We denote the set of all strongly asymptotic fixed point of T by F^(T).

A mapping T: C → C is said to be nonexpansive, if

||Tx-Ty||≤||x-y||,∀x,y∈C.

A mapping T: C → C is said to be relatively nonexpansive if F (T) ≠ Ø, F(T)=F̃(T), and

ϕ(u,Tx)≤ϕ(u,x),∀u∈F(T),x∈C

A mapping T: C → C is said to be weak relatively nonexpansive if F(T) ≠ Ø, F(T)=F^(T), and

ϕ(u,Tx)≤ϕ(u,x),∀u∈F(T),x∈C

A mapping T: C → C is said to be quasi-ϕ-nonexpansive, if F(T) ≠ Ø and

ϕ(u,Tx)≤ϕ(u,x),∀x∈C,∀u∈F(T)

A mapping T: C → C is said to be quasi-ϕ-asymptotically nonexpansive, if there exists some real sequence {kn } with kn≥ 1 and kn → 1 and F (T) ≠ ∅ such that

ϕ(u,Tnx)≤knϕ(u,x),∀n≥1,x∈C,u∈F(T)

(1.7)

A mapping T: C → C is said to be closed, if for any sequence {xn } ⊂C with xn → x and Txn → y, then Tx = y.

Motivated by the above, the purpose of this article is to introduce the concept of total quasi-ϕ-asymptotically nonexpansive mapping which contains many kinds of mappings as its special cases and we prove a strong convergence theorem by using a hybrid method for finding a common element of the set of solutions for a generalized mixed equilibrium problems, the set of fixed points of a family of total quasi- ϕ-asymptotically nonexpansive mappings in uniformly smooth and strictly convex Banach space with the Kadec-Klee property. The results presented in the paper improve and extend some recent results.

2 Preliminaries

Throughout this article, we assume that all the Banach spaces are real. We denote by ℕ and ℝ the sets of positive integers and real numbers, respectively. Let E be a Banach space and let E* be the topological dual of E. For all x∈E and x*∈E* , we denote by 〈x, x*〉 the value of x* at x. The mapping J: E → 2E*defined by

J(x)={x*∈E*:〈x,x*〉=||x||2=||x*||2},x∈E,

(2.1)

is normalized duality mapping. We denote the weak convergence and the strong convergence of a sequence {xn } to x by xn⇀x and xn → x, respectively.

A Banach spaces E is said to be strictly convex if ||x+y||2<1 for x, y∈S(E) = {z∈E: ||z|| = 1} with x ≠ y. It is said to be uniformly convex if for any given ε∈ (0, 2], there exists δ > 0 such that ||x+y||2<1-δ for x, y∈S(E) with ||x - y|| ≥ ε. E is said to have the Kadec-Klee property, if for any sequence {xn } ⊂E such that xn⇀x∈E and ||xn || → ||x||, then xn → x.

Define f: S(E) × S(E) ×ℝ \ {0} →ℝ by

f(x,y,t)=∥x+ty∥-∥x∥t

for x, y∈S(E) and t∈ℝ \ {0}. A norm of E is said to be Gâteaux differentiable if limt→ 0f(x, y, t) has a limit for each x, y∈S(E). In this case, E is said to be smooth. We know that if E is smooth, strictly convex, and reflexive, then the duality mapping J is single valued, one to one, and onto. In this case, the inverse mapping J-1 coincides with the duality mapping J* on E*. See [14] for more details.

Remark 2.1 If E is a reflexive and strictly convex Banach space, then J-1 is hemi-continuous, i.e., J-1 is norm-weak-continuous.

Let {Cn } be a sequence of nonempty closed convex subset of a reflexive Banach space E. We define two subsets s - LinCn and w - LsnCn as follows: x∈s - LinCn if and only if there exists {xn } ⊂E such that {xn } converges strongly to x and that xn∈Cn for all n∈ℕ. Similarly, y∈w - LsnCn if and only if there exists a subsequence {Cni} of {Cn } and a sequence {yi } ⊂E such that {yi } converges weakly to y and yi∈Cni for all i∈ℕ. We define the Mosco convergence [15] of {Cn } as follows: If C0 = s - LinCn = w - LsnCn , then {Cn } is said to be convergent to C0 in the sense of Mosco and we write C0 = M - limn→∞Cn .

Lemma 2.2. Let C be a nonempty closed convex subsets of a smooth, strictly convex, and reflexive Banach spaces E, let x∈E and z∈C. Then the following conclusions hold:

(a)

z = ∏ Cx⇔〈y - z, Jx - jz〉 ≤ 0, ∀y∈C.

(b)

For x, y∈E, ϕ (x, y) = 0 if and only if x = y.

The following theorem proved by Tsukada [19] plays an important role in our results.

Theorem 2.1. Let E be a smooth, reflexive, and strictly convex Banach spaces having the Kadec-Klee property. Let {Kn } be a sequence of nonempty closed convex subsets of E. If K0 = M - limn→∞Kn exists and is nonempty, then PKnx converges strongly to PK0x for each x∈C.

Theorem 2.1 is still valid if we replace the metric projections with the generalized pro-jections as follows:

Theorem 2.2 Let E be a smooth, reflexive, and strictly convex Banach spaces having the Kadec-Klee property. Let {Kn } be a sequence of nonempty closed convex subsets of E. If K0 = M - limn→∞Kn exists and is nonempty, then ∏Knx converges strongly to ∏K0x for each x∈C.

For solving the equilibrium problem for bifunction F: C × C → ℝ, let us assume that F satisfies the following conditions:

(A1) F (x, x) = 0 for all x∈C;

(A2) F is monotone, i.e., F (x, y) + F (y, x) ≤ 0 for all x, y∈C;

(A3) for each x, y, z∈C,

lim supt↓0F(tz+(1-t)x,y)≤F(x,y);

(A4) for each x∈C, y α F (x, y) is a convex and lower semicontinuous.

If an equilibrium bifunction F: C × C → R satisfies conditions (A1)-(A4), then we have the following results.

Since 픉 is nonempty, Cn is a nonempty closed convex subset of E and thus ΠCn exists for every n∈ℕ. Hence {xn } is well defined. Also, since {Cn } is a decreasing sequence of closed convex subsets of C such that C0=⋂n=1∞Cn is nonempty. Therefore, we have

From (2.2) it yields supλ∈Λ(||xn+1|| - ||uλ, n||)2→ 0. Since ||xn+1|| → ||x*||, we have

∥uλ,n∥→∥x*∥(n→∞),∀λ∈Λ

(3.6)

Hence we have

||Juλ,n||→||Jx*||n→∞,∀λ∈Λ

(3.7)

This implies that {Juλ, n } is uniformly bounded in E*. Since E is reflexive, and so is E*. We can assume that Juλ, n⇀f0∈E*. In view of the reflexive of E, we see that J(E) = E*. Hence there exists p∈E such that Jp = f0. Since

i.e., x* = p. This implies that f0 = Jx*, and so Juλ, n⇀Jx*, ∀λ∈ Λ. It follows from (3.7) and the Kadec-Klee property of E* that Juλ, n→ Jx*(n → ∞). Note that J-1: E* → E is hemi-continuous, it yields that uλ, n⇀x*. In view of (3.6) and the Kadec-Klee property of E, we have

This implies that {Jyλ, n} is bounded in E*. Since E is reflexive, and so is E*. we can assume that Jyλ, n⇀g0∈E*. In view of the reflexive of E, we see that J(E) = E*. Hence there exists y∈E such that Jy = g0. Since

i.e., x* = y. This implies that g0 = Jx*, and so Jyλ, n⇀Jx*. It follows from (3.19) and the Kadec-Klee property of E* that Jyn (λ) → Jx*(n → ∞). Note that J-1: E* → E is hemicontinuous, it yields that yλ, n⇀x*. It follows from (3.18) and the Kadec-Klee property of E that

limn→∞yλ,n=x*.

(3.20)

Since uλ, n→ x*, from (3.20), we have

limn→∞||uλ,n-yλ,n||=0.

(3.21)

Since J is uniformly norm-to-norm continuous on bounded sets, from (3.21), we have

limn→∞||Juλ,n-Jyλ,n||=0.

(3.22)

From rλ, n≥ a, we have

limn→∞||Juλ,n-Jyλ,n||rλ,n=0.

(3.23)

By uλ,n=Wrλ,nyλ,n, we have

G(uλ,n,y)+1rλ,n〈y-uλ,n,Juλ,n-Jyλ,n〉≥0,∀y∈C.

(3.24)

From condition (A2), we have

1rλ,n〈y-uλ,n,Juλ,n-Jyλ,n〉≥-G(uλ,n,y)≥G(y,uλ,n),∀y∈C.

(3.25)

Since G(x,·) is convex and lower semicontinuous, it is also weakly lower semicontinuous, letting n → ∞, we have from (3.25) and (A4) that

where ξn=(1-αn)(vnsupu∈픉ζ(ϕ(u,xn))+μn), {αn } is a sequence in 0[1] such that lim infn→∞ αn(1 - αn ) > 0, lim inf n→∞αn< 1 and {rλ, n} ⊂ [a, ∞) for some a > 0, then {xn } converge strongly to some point x* in F.

Declarations

Acknowledgements

This study was supported by Scientific Research Fund of Sichuan Provincial Education Department (11ZB146)(2012), the natural Science Foundation of Yibin University (2011Z08) and Scientific Research Fund of Science Technology Department of Sichuan Province (2011JYZ011).

Competing interests

The authors declare that they have no competing interests.

Authors' contributions

All the authors contributed equally to the writing of the present article. All authors read and approved the final manuscript.

Authors’ Affiliations

(1)

Department of Mathematics, Yibin university

(2)

Department of Mathematics, College of Statistics and Mathematics, Yunnan University of Finance and Economics

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